please make sure algorithm is in O(lgn) time and a full proof explaining why it

Question

please make sure algorithm is in O(lgn) time and a full proof explaining why it works (loop invariant, etc).

4. (20 pts) Let A and B be arrays of integers. Each array contains n elements, and each array is in sorted order (ascending). A and B do not share any elements in common. Give a O(lg n)-time algorithm which finds the median of A U B and prove that it is correct. This algorithm will thus find the median of the 2n elements that would result from putting A and B together into one array. (Note: define the median to be the average of the two middle values of a list with an even number of elements.)

Explanation / Answer

This method works by first getting medians of the two sorted arrays and then comparing them.

Let ar1 and ar2 be the input arrays.

Algorithm:

1) Calculate the medians m1 and m2 of the input arrays ar1[]
and ar2[] respectively.
2) If m1 and m2 both are equal then we are done.
return m1 (or m2)
3) If m1 is greater than m2, then median is present in one
of the below two subarrays.
a) From first element of ar1 to m1 (ar1[0 to n/2])
b) From m2 to last element of ar2 (ar2[n/2 to n-1])
4) If m2 is greater than m1, then median is present in one
of the below two subarrays.
a) From m1 to last element of ar1 (ar1[n/2 to n-1])
b) From first element of ar2 to m2 (ar2[0 to n/2])
5) Repeat the above process until size of both the subarrays
becomes 2.
6) If size of the two arrays is 2 then use below formula to get
the median.
Median = (max(ar1[0], ar2[0]) + min(ar1[1], ar2[1]))/2

Example:

ar1[] = {1, 12, 15, 26, 38}
ar2[] = {2, 13, 17, 30, 45}
For above two arrays m1 = 15 and m2 = 17

For the above ar1[] and ar2[], m1 is smaller than m2. So median is present in one of the following two subarrays.

[15, 26, 38] and [2, 13, 17]
Let us repeat the process for above two subarrays:

m1 = 26 m2 = 13.
m1 is greater than m2. So the subarrays become

[15, 26] and [13, 17]
Now size is 2, so median = (max(ar1[0], ar2[0]) + min(ar1[1], ar2[1]))/2
= (max(15, 13) + min(26, 17))/2
= (15 + 17)/2
= 16

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